Abstract

Space-based inflatable technology is of current interest to NASA and DOD, and in particular to the Air Force and Phillips Laboratory. Potentially large gains in lowering launch costs, through reductions in structure mass and volume, are driving this activity. Diverse groups are researching and developing this technology for radio and radar antennae, optical telescopes, and solar power and propulsion applications. Regardless of the use, one common requirement for successful application is the accuracy of the inflated surface shape. The work reported here concerns the shape control of an inflated thin circular disk through use of a nonlinear finite element analysis. First, a review of the important associated Hencky problem is given. Then we discuss a shape modification, achieved through enforced boundary displacements, which resulted in moving the inflated shape towards a desired parabolic profile. Minimization of the figure error is discussed and conclusions are drawn.

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References

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  1. Jenkins, C.H., and Leonard, J.W., "Nonlinear Dynamic Response of Membranes: State of the Art," Appl. Mech. Rev. 44, 319-328 (1991).
    [CrossRef]
  2. Jenkins, C.H., "Nonlinear Dynamic Response of Membranes: State of the Art -- Update," Appl. Mech. Rev. 49 (10), S41-S48 (1996).
    [CrossRef]
  3. Hencky, H., "Uber den Spannungszustand in kreisrunden Platten," Z. Math. Phys. 63, 311-317 (1915).
  4. Foeppl, A., "Vorlesungen uber technische Mechanik," B.G. Teubner, Bd. 5., p. 132, Leipzig, Germany (1907).
  5. von Karman, T., Festigkeitsproblem im Naschinenbau, Encyk. D. Math. Wiss. IV, 311-385 (1910).
  6. Stevens, H.H., "Behavior of circular membranes stretched above the elastic limit by air pressure," Exp. Stress Anal. 2, 139-146 (1944).
  7. Chien, W.Z., "Asymptotic behavior of a thin clamped plate under uniform normal pressure at very large deflection," Sci. Rep. Natn. Tsing Hua Univ. A5, 71-94 (1948).
  8. Kao, R., and Perrone, N., Large deflections of axisymmetric circular membranes, Int. J. Solids Struct. 7, 1601-1612 (1971).
    [CrossRef]
  9. Cambell, J.D., On the theory of initially tensioned circular membranes subjected to uniform pressure, Q. J. Mech. Appl. Math. 9, 84-93 (1956).
    [CrossRef]
  10. Dickey, R.W., The plane circular elastic surface under normal pressure, Arch. Ration. Mech. Anal. 26, 219-236 (1967).
    [CrossRef]
  11. Weil, N.A., and Newmark, N.M., Large plastic deformations of circular membranes, J. Appl. Mech. 22, 533-538 (1955).
  12. Kao, R., and Perrone, N., Large deflections of flat arbitrary membranes, Comput. Struct. 2, 535-546 (1972).
    [CrossRef]
  13. Shaw, F.S., and Perrone, N., A numerical solution for the non-linear deflection of membranes, J. Appl. Mech. 21, 117-128 (1954).
  14. Schmidt, R., and DaDeppo, D.A., A new approach to the analysis of shells, plates, and membranes with finite deflections, Int. J. Non-Linear Mech. 9, 409-419 (1974).
    [CrossRef]
  15. Schmidt, R., On Berger's method in the non-linear theory of plates, J. Appl. Mech. 41, 521-523 (1974).
    [CrossRef]
  16. Storakers, B., Small deflections of linear elastic circular membranes under lateral pressure, J. Appl. Mech. 50, 735-739 (1983).
    [CrossRef]
  17. Weinitschke, H.J., On axisymmetric deformations of nonlinear elastic membranes, Mech. Today 5, 523-542, Pergamon Press, Oxford (1980).
  18. Weinitschke, H.J., On finite displacements of circular elastic membranes, Math. Method Appl. Sci. 9, 76-98 (1987).
    [CrossRef]
  19. Weinitschke, H.J., Stable and unstable axisymmetric solutions for membranes of revolution, Appl. Mech. Rev. 42, S289-S294 (1989).
    [CrossRef]
  20. Ciarlet, P.G., A justification of the von Krmn equations, Arch. Ration. Mech. Anal. 73, 349-389 (1980).
    [CrossRef]
  21. Pujara, and Lardner, T.J., Deformations of elastic membranes--effect of different constitutive relations, Z. Angew. Math. Phys. 29, 315-327 (1978).
    [CrossRef]
  22. Thomas, M., and Veal, G. (1984), Highly accurate inflatable reflectors, AFRPL TR-84-021.
  23. Murphy, L.M., Stretched-membrane heliostat technology, J. Solar Energy Eng. 108, 230-238 (1986).
    [CrossRef]
  24. Murphy, L.M., Moderate axisymmetric deformations of optical membrane surfaces, J. Solar Energy Eng. 109, 111-120 (1987).
    [CrossRef]
  25. Palisoc, A., and Thomas, M., A comparison of the performance of seamed and unseamed inflatable concentrators, Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf., 2, 855-864 (1995).
  26. Basart, J. P., Mandayam, S.A., and Burns, J.O., An inflatable antenna for space-based low-frequency radio astronomy, Proc. Space 94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).
  27. Cassapakis, C., and Thomas, M., Inflatable structures technology development overview, AIAA 95-3738 (1995).
  28. Chow, P.Y., Construction of pressurized, self-supporting membrane structure on the moon, J. Aerospace Eng. 5, 274-281 (1992).
    [CrossRef]
  29. Grossman, G., and Williams, G., Inflatable concentrators for solar propulsion and dynamic space power, J. Solar Energy Eng. 112, 229-236 (1990).
    [CrossRef]
  30. Malone, P.K., and Williams, G. T., A lightweight inflatable solar array, Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).
  31. Nowak, P.S., Sadeh, W.Z., and Janakus, J., Feasibility study of inflatable structures for lunar base, J. Spacecraft Rockets 31, 453-457 (1994).
    [CrossRef]
  32. Rogers, C.A., Stultzman, W.L., Campbell, T.G., and Hedgepeth, J.M., Technology assessment and development of large deployment antennas, J. Aerospace Engineering 6(1), 34-54 (1993).
    [CrossRef]
  33. Sadeh, W.Z., and Criswell, M.E., A generic inflatable structure for a lunar/Martian base, Space IV, Proc. Space '94, Albuquerque, ASCE, 1146-1156 (1994).
  34. Hedgepeth, J.M., Accuracy potentials for large space antenna reflectors with passive structures, J. Spacecraft 19(3), 211-217 (1982).
    [CrossRef]
  35. Vaughn, H., Pressurizing a prestretched membrane to form a paraboloid, Int. J. Eng. Sci. 18, 99-107 (1980).
    [CrossRef]
  36. Hart-Smith, L.J., and Crisp, J.D.C., Large elastic deformations of thin rubber membranes, Int. J. Eng. Sci. 5, 1-24 (1967).
    [CrossRef]
  37. Natori, M., Shibayama, Y., and Sekine, K., Active accuracy adjustment of reflectors through the change of element boundary, AIAA 89-1332 (1989).
  38. Jenkins, C.H., Marker, D.K., and Wilkes, J.M., Improved surface accuracy of precision membrane reflectors through adaptive rim control, AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).
  39. Jenkins, C.H., Wilkes, J.M., and Marker, D.K., Surface accuracy of precision membrane reflectors, Space 98, Albuquerque, NM (to appear) (1998b).
    [CrossRef]

Other (39)

Jenkins, C.H., and Leonard, J.W., "Nonlinear Dynamic Response of Membranes: State of the Art," Appl. Mech. Rev. 44, 319-328 (1991).
[CrossRef]

Jenkins, C.H., "Nonlinear Dynamic Response of Membranes: State of the Art -- Update," Appl. Mech. Rev. 49 (10), S41-S48 (1996).
[CrossRef]

Hencky, H., "Uber den Spannungszustand in kreisrunden Platten," Z. Math. Phys. 63, 311-317 (1915).

Foeppl, A., "Vorlesungen uber technische Mechanik," B.G. Teubner, Bd. 5., p. 132, Leipzig, Germany (1907).

von Karman, T., Festigkeitsproblem im Naschinenbau, Encyk. D. Math. Wiss. IV, 311-385 (1910).

Stevens, H.H., "Behavior of circular membranes stretched above the elastic limit by air pressure," Exp. Stress Anal. 2, 139-146 (1944).

Chien, W.Z., "Asymptotic behavior of a thin clamped plate under uniform normal pressure at very large deflection," Sci. Rep. Natn. Tsing Hua Univ. A5, 71-94 (1948).

Kao, R., and Perrone, N., Large deflections of axisymmetric circular membranes, Int. J. Solids Struct. 7, 1601-1612 (1971).
[CrossRef]

Cambell, J.D., On the theory of initially tensioned circular membranes subjected to uniform pressure, Q. J. Mech. Appl. Math. 9, 84-93 (1956).
[CrossRef]

Dickey, R.W., The plane circular elastic surface under normal pressure, Arch. Ration. Mech. Anal. 26, 219-236 (1967).
[CrossRef]

Weil, N.A., and Newmark, N.M., Large plastic deformations of circular membranes, J. Appl. Mech. 22, 533-538 (1955).

Kao, R., and Perrone, N., Large deflections of flat arbitrary membranes, Comput. Struct. 2, 535-546 (1972).
[CrossRef]

Shaw, F.S., and Perrone, N., A numerical solution for the non-linear deflection of membranes, J. Appl. Mech. 21, 117-128 (1954).

Schmidt, R., and DaDeppo, D.A., A new approach to the analysis of shells, plates, and membranes with finite deflections, Int. J. Non-Linear Mech. 9, 409-419 (1974).
[CrossRef]

Schmidt, R., On Berger's method in the non-linear theory of plates, J. Appl. Mech. 41, 521-523 (1974).
[CrossRef]

Storakers, B., Small deflections of linear elastic circular membranes under lateral pressure, J. Appl. Mech. 50, 735-739 (1983).
[CrossRef]

Weinitschke, H.J., On axisymmetric deformations of nonlinear elastic membranes, Mech. Today 5, 523-542, Pergamon Press, Oxford (1980).

Weinitschke, H.J., On finite displacements of circular elastic membranes, Math. Method Appl. Sci. 9, 76-98 (1987).
[CrossRef]

Weinitschke, H.J., Stable and unstable axisymmetric solutions for membranes of revolution, Appl. Mech. Rev. 42, S289-S294 (1989).
[CrossRef]

Ciarlet, P.G., A justification of the von Krmn equations, Arch. Ration. Mech. Anal. 73, 349-389 (1980).
[CrossRef]

Pujara, and Lardner, T.J., Deformations of elastic membranes--effect of different constitutive relations, Z. Angew. Math. Phys. 29, 315-327 (1978).
[CrossRef]

Thomas, M., and Veal, G. (1984), Highly accurate inflatable reflectors, AFRPL TR-84-021.

Murphy, L.M., Stretched-membrane heliostat technology, J. Solar Energy Eng. 108, 230-238 (1986).
[CrossRef]

Murphy, L.M., Moderate axisymmetric deformations of optical membrane surfaces, J. Solar Energy Eng. 109, 111-120 (1987).
[CrossRef]

Palisoc, A., and Thomas, M., A comparison of the performance of seamed and unseamed inflatable concentrators, Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf., 2, 855-864 (1995).

Basart, J. P., Mandayam, S.A., and Burns, J.O., An inflatable antenna for space-based low-frequency radio astronomy, Proc. Space 94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).

Cassapakis, C., and Thomas, M., Inflatable structures technology development overview, AIAA 95-3738 (1995).

Chow, P.Y., Construction of pressurized, self-supporting membrane structure on the moon, J. Aerospace Eng. 5, 274-281 (1992).
[CrossRef]

Grossman, G., and Williams, G., Inflatable concentrators for solar propulsion and dynamic space power, J. Solar Energy Eng. 112, 229-236 (1990).
[CrossRef]

Malone, P.K., and Williams, G. T., A lightweight inflatable solar array, Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).

Nowak, P.S., Sadeh, W.Z., and Janakus, J., Feasibility study of inflatable structures for lunar base, J. Spacecraft Rockets 31, 453-457 (1994).
[CrossRef]

Rogers, C.A., Stultzman, W.L., Campbell, T.G., and Hedgepeth, J.M., Technology assessment and development of large deployment antennas, J. Aerospace Engineering 6(1), 34-54 (1993).
[CrossRef]

Sadeh, W.Z., and Criswell, M.E., A generic inflatable structure for a lunar/Martian base, Space IV, Proc. Space '94, Albuquerque, ASCE, 1146-1156 (1994).

Hedgepeth, J.M., Accuracy potentials for large space antenna reflectors with passive structures, J. Spacecraft 19(3), 211-217 (1982).
[CrossRef]

Vaughn, H., Pressurizing a prestretched membrane to form a paraboloid, Int. J. Eng. Sci. 18, 99-107 (1980).
[CrossRef]

Hart-Smith, L.J., and Crisp, J.D.C., Large elastic deformations of thin rubber membranes, Int. J. Eng. Sci. 5, 1-24 (1967).
[CrossRef]

Natori, M., Shibayama, Y., and Sekine, K., Active accuracy adjustment of reflectors through the change of element boundary, AIAA 89-1332 (1989).

Jenkins, C.H., Marker, D.K., and Wilkes, J.M., Improved surface accuracy of precision membrane reflectors through adaptive rim control, AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).

Jenkins, C.H., Wilkes, J.M., and Marker, D.K., Surface accuracy of precision membrane reflectors, Space 98, Albuquerque, NM (to appear) (1998b).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Definition Sketch

Fig.2.
Fig.2.

Inflated Membrane (3-D view)

Fig. 3.
Fig. 3.

Comparison Measurements Fig.

Fig. 4.
Fig. 4.

Radial Displacements

Fig. 5.
Fig. 5.

Model Vs Parabola

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ε r = du dr + 1 2 ( dw dr ) 2
ε θ = u r
D ( d 2 d r 2 + 1 r ) 2 w h r d dr ( d Φ dr dw dr ) = p
r d dr ( d 2 Φ d r 2 + 1 r d Φ dr ) + E 2 ( dw dr ) 2 = 0
σ r = 1 r d Φ dr , σ θ = d 2 Φ d r 2
k = ( 2 r a p Eh ) 1 3
w = w 0 ( 1 + α 2 r 2 a 2 + α 4 r 4 a 4 )
w 0 = 0.626 a ( pa Eh ) 1 3
M r = D ( d 2 w d r 2 + ν r dw dr )
ε θ = 0 σ θ ν σ r = 0 d 2 Φ d r 2 ν r d Φ dr
α 2 = 6 + 2 ν 5 + ν
α 4 = 1 + ν 5 + ν
w b = w 0 ( 1 1.259 r 2 a 2 + 0.245 r 4 a 4 )
w m = w 0 ( 1 0.899 r 2 a 2 0.101 r 4 a 4 )
w p = w 0 ( 1 r 2 a 2 )
k m = Eh 1 ν 2
ε θ = 0 σ θ = ν σ r
ν ( r = a ) = σ θ σ r
k m ( r = 0 ) k m ( r = a ) = 1 [ v ( a ) ] 2 1 [ ν ( 0 ) ] 2 > 1

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